{
  "abstract": "We present a computational framework that combines machine-learning-based discovery with rigorous computer-assisted numerical certification and formal verification for strongly nonlinear partial differential equations. We demonstrate the approach for periodic gravity-capillary traveling water waves. Physics-informed neural networks are used as a search mechanism, with the wave profile, wave speed and, in continuation experiments, surface tension learned directly from the governing equations without training data. Through a staged training and refinement procedure, the neural networks achieve near-machine-precision residuals of the governing equations, even in strongly nonlinear regimes. A single formulation recovers high-amplitude Stokes waves, Crapper waves and Wilton ripples. Continuation from a high-amplitude Wilton ripple in surface tension reveals new high-amplitude candidate solutions. We freeze one candidate as an explicit finite representation and rigorously certify a uniform bound on its governing-equation residual using interval arithmetic, accounting for floating-point roundoff, Fourier truncation, aliasing and grid-to-continuum errors. This independently validates the neural-network accuracy. We formalize the certification argument in Lean, which machine-checks the deductions from certified numerical bounds to the residual theorem. Both the interval certification and formalization were developed with substantial assistance from a large language model, while the validity of the result remains independent of the language model: the numerical bounds are rigorously enclosed by interval arithmetic and the subsequent mathematical deductions are checked by the Lean kernel. Together, these steps provide a pipeline from neural discovery to rigorous numerical certification and machine-checked verification with auditable mathematical guarantees.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07395",
  "author_total": 3,
  "authors": [
    {
      "name": "Michal Shavit"
    },
    {
      "name": "Yongji Wang"
    },
    {
      "name": "Tristan Buckmaster"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/atxd9d403sqdk.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07395",
  "fulltext_url": null,
  "id": "atxd9d403sqdk",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07395",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "physics.flu-dyn"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07395",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/atxd9d403sqdk.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:30Z",
      "native_id": "2611.07395",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07395"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.flu-dyn",
      "subfield": "fluid-dynamics"
    }
  ],
  "title": "Rigorously Certified Neural-Network Computation of Nonlinear Water Waves at Near-Machine Precision",
  "title_text": "Rigorously Certified Neural-Network Computation of Nonlinear Water Waves at Near-Machine Precision",
  "type": "preprint",
  "updated_at": "2026-11-08T18:20:59Z",
  "url": "https://research.draconitic.com/paper/atxd9d403sqdk",
  "venue": "arXiv",
  "versions": []
}